Trigonometric functions and the unit circle

Question

I have been trying to understand the concepts that I didn’t fully grasp in high school calculus level lately, and it has been in the area of trig where I have the most trouble. So, as I have understood, the unit circle lets mathematicians define the functions sine cosine and tangent on a way that allows them to have the value of any real input, which is interpreted as the angle created by traveling along the circle counterclockwise with the X axis. I understand how to represent any point in the first quadrant using the cos and sin but I am not sure if it makes sense for the rest of the circle. If we take any other point in the circle, belonging to another quadrant, then the cosine or sine of that angle bigger than 90 degrees is not defined clearly, since what we should do is take actually the reference angle in the first quadrant and then build our triangle to get the correct ratios multiplied by its corresponding sign depending on where the point ended up due to the input angle initially. So this is what tricks me, I am not “satisfied” by accepting that the ways the trig function is defined for values greater than 90 and below 360 are actually the values of that same trig function but taking a value between 0 and 90 and scaling by +1/-1 depending on where the point is (which is seen by moving counterclockwise until reaching the angle given ) It discomforts me accepting this without confirmation since I feel like then the trig functions could have been more easily defined by just saying that it’s actual domain is 0-90 degrees because at the end that’s really what it computes by differing on a sign based on the quadrant. It feels like any input above 90 actually doesn’t get accepted directly and has to be correctly reduced into the 0-90 interval, taking into account the sign as well by its unit circle reasoning behind.

Answer 1

It can be more or less difficult to rigorously define the trigonometric functions (you would have to define properly what is an angle, which is not trivial as well). Personally I use the unit circle as a mnemonic but not as a definition.

If you want a clear and unambiguous definition, the best (from my opnion) is probably to define $\cos$ and $\sin$ functions with power series :

$$ \begin{aligned} &\cos (x)=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\cdots=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !} \\ &\sin (x)=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\cdots=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{(2 n+1) !} \end{aligned} $$

But I understand such definitions can feel unintuitive. Where are the geometric properties here ? Why do we have infinite sums ?... You can derive from these formulas all the familiar properties you know but it wont be as obvious as with a direct geometric definition.

Other possibilities are :

• indeed using a rigorous geometric definition, but you will need to define properly every object (angle, length...), requiring some level in linear and bilinear algebra.
• using differential equations $y'' + y = 0$ and proper initial conditions
• using the real and imaginary part of the complex exponential etc...

All of these techniques are more or less linked but they all require a level above what is expected from an average high-schooler. So if you felt that it wasn't properly defined in high school, don't worry it's totally normal.

Answer 2

So, as I have understood, the unit circle lets mathematicians define the functions sine cosine and tangent on a way that allows them to have the value of any real input, which is interpreted as the angle created by traveling along the circle counterclockwise with the X axis.

This is wrong. In Calculus (aka Real Analysis), the domain of the sine and cosine functions is the Real number associated with a specific arc length. Since the circumference of the unit circle happens to be $(2\pi)$, and since (in Analytical Geometry or Trigonometry) this translates to $(360^\circ)$, students new to Calculus are taught about radians, which is a very confusing and ambiguous term.

Such students are taught that $(2\pi)$ radians equals $(360^\circ)$, and so the (new) Calculus student continues to regard the domain of the sine and cosine functions as angles, measured in radians.

Then, as you progress in Calculus, you start encountering Math problems that render the above paragraph nonsensical. One example is the Taylor series for the sine and cosine functions. Another (classic) example is

$$\int_0^1 \frac{dt}{1 + t^2} = \text{arctan}(1) - \text{arctan}(0). \tag1 $$

At this point the student becomes confused, because the LHS of (1) must evaluate to a Real number, while the student's initial understanding of radians causes the RHS to evaluate as the difference of two angles.

Then, the student realizes that they have to abandon the notion that a radian represents the measurement of an angle. Instead, the student learns to regard the radian as a Real number. That is $(2\pi)$ radians equals the circumference of the unit circle.

With this understanding, the student realizes that the RHS of (1) above evaluates to $(\pi/4) - 0 = \pi/4,~$ which is a Real number.